Divine Proportions: Rational Trigonometry to Universal Geometry

N. J. Wildberger

Divine Proportions is not a textbook designed for a wide audience. It’s more of a proof-of-concept, the development of the author’s unconventional approach to trigonometry and Euclidean geometry “intended for a mathematically mature audience.” However, N. J. Wildberger believes that his book “introduces a remarkable new approach to trigonometry and Euclidean geometry, with dramatic implications for mathematics teaching, industrial applications and the directions of mathematical research in geometry. […] Teachers and students will benefit from a simpler and cleaner theory… This book… provides the mathematical foundation for a dynamic and elegant new approach to teaching trigonometry and geometry.”

At the most basic level Wildberger replaces the concepts of distance and (ordinary) angle measure with quantities he calls quadrance and spread:

quadrance = (distance)2

spread = (sin(angle))2

However, since quadrance and spread are the basic quantities in the book, they are not defined from distance or from ordinary angle measure. And they certainly do not rely for their definition on transcendental functions such as sine or cosine.

Instead, Wildberger assumes that his points are ordered pairs of numbers [x,y]. Then the quadrance between two points A1 = [x1,y1] and A2 = [x2,y2] is defined to be

Q(A1,A2) = (x1–x2)2 + (y1–y2)2.

Similarly, given two lines l1 and l2 specified by equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 , then the spread between the two lines is defined to be

.

The primary advantages of quadrance and spread over distance and ordinary angle measure is that they are quadratic functions of the coefficients, and they do not involve square roots or transcendental functions. Hence, in particular, these quantities can be defined when the numbers under consideration are from fields more general than the fields of real or rational numbers. This generalization leads to what Wildberger labels as Universal Geometry.

Wildberger then proceeds to develop the basic (and several advanced) topics in Euclidean geometry and Universal Geometry with no significant reference to distance and absolutely no reference to ordinary angle measure. Because of the quadratic nature of quadrance and spread this development is done with elementary (though often messy) arithmetic and algebra. This does not lead to new results, but to alternate formulations of classical results. In particular, Wildberger labels the “five main laws of rational trigonometry” as the

triple quad formula,

Pythagoras’ theorem (for quadrances),

spread law,

cross law,

triple spread formula.

When reformulated using classical trigonometric notions, these laws correspond to the

additive formula for distance between three collinear points,

Pythagoras’ theorem (for distances),

Law of Sines,

Law of Cosines,

sum of angles of a triangle equals 180°.

Given that the book is intended for a mathematically mature audience, the development of the material, the calculations, and the accompanying discussions are well organized, clear, and well-written. I found no flaws in the mathematics (although I did not read every argument in depth). The book presents an interesting and creative approach to the development of trigonometry and geometry. I found Wildberger’s Universal Geometry, especially when applied to finite fields, particularly novel. This could provide fertile ground for further investigation.

However, I have reservations about the superiority of Wildberger’s methods over the classical theory. He believes his methods constitute a “simpler and more logical approach to trigonometry and geometry,” and should ultimately replace the classical approach in schools and colleges as the primary development of these subjects. I’m not convinced of the wisdom of attempting such a change.

The intuitive geometric clarity of distance and angle measure does not carry over to quadrance and spread. Moreover, additivity for distance between collinear points and for angle measure between adjacent angles is natural, expected, and useful in the real world; this property does not hold for quadrance and spread. There have to be extremely compelling reasons to replace the intuitively appealing, additive geometric quantities of distance and angle measure with less-intuitive, non-additive quantities.

As one justification for this replacement Wildberger claims that computations with quadrance and spread are simpler than with distance and angle measure. However, the increased simplicity comes not so much from reducing the number of computational steps needed to solve a problem (indeed, the necessary algebraic steps can be extensive in rational trigonometry) but from changing the nature of the computations: reliance on quadratic algebra and the elimination of transcendental functions. But would these changes make mastery of trigonometry and geometry easier for our students? The introduction of the sine and cosine is difficult for many, but not because these functions are transcendental but because they are unfamiliar and require the use of many sophisticated identities. But the spread of two lines is also unfamiliar, and many of the rational trigonometry formulas are as complicated as their classical counterparts.

Individual evaluations of the importance of Wildberger’s new approach to trigonometry and geometry will probably depend on one’s viewpoints on the foundations of mathematics. Certainly Wildberger’s views on foundational issues have strongly motivated and influenced the directions taken in Divine Proportions. Wildberger believes there are many logical difficulties with “infinite set theory” — in his paper “Set Theory: Should You Believe?” Wildberger states “It is not clear that there are any sets that are not finite…” This motivates some rather non-standard definitions in Divine Proportions. For example, a line is not a set of points; it is a “3-proportion” with individual points either “on” or not “on” the line.

Wildberger’s dislike of “infinite set theory” leads to equally critical views concerning analysis and, in particular, the real number system. Consequently, as stated in the Introduction to Divine Proportions, his “new theory unites the three core areas of mathematics — geometry, number theory and algebra — and expels analysis and infinite processes from the foundations of the subject.” Anyone sharing Wildberger’s unconventional views will hail “rational trigonometry” for its lack of dependence on the field of real numbers and its avoidance of transcendental functions. However, if you do not share these critical views of analysis — I do not — then the claim of superiority of “rational trigonometry” over classical trigonometry loses much of its attraction.

Divine Proportions is unquestionably a valuable addition to the mathematics literature. It carefully develops a thought provoking, clever, and useful alternate approach to trigonometry and Euclidean geometry. It would not be surprising if some of its methods ultimately seep into the standard development of these subjects. However, unless there is an unexpected shift in the accepted views of the foundations of mathematics, there is not a strong case for rational trigonometry to replace the classical theory.

William Barker is the Isaac Henry Wing Professor of Mathematics at Bowdoin College. He received his Ph.D. at M.I.T. in 1973, writing a thesis under the guidance of Prof. Sigurdur Helgason in analysis on Lie groups. Barker was subsequently a John Wesley Young Research Instructor at Dartmouth College for two years, joining the Bowdoin faculty in 1975. His most recent work has been an undergraduate geometry textbook, Continuous Symmetry: From Euclid to Klein, co-authored with Roger Howe of Yale University. A second volume is currently under development. Barker has also been active with the MAA, taking part in the production of CRAFTY’s Curriculum Foundations Project and the CUPM’s Curriculum Guide 2004. He can be reached at barker@bowdoin.edu.